Exercise 17A: Exponents and Logarithms | 练习 17A：指数与对数

📚 Exercise 17A: Exponents and Logarithms | 练习 17A：指数与对数

Exponents and logarithms form the backbone of many IB Mathematics problems, from solving equations to modelling real‑world phenomena. This article unpacks the key concepts and techniques you need to master Exercise 17A, offering clear explanations and worked examples that mirror typical exam questions.

指数与对数是 IB 数学中许多题目的支柱，从解方程到建模现实世界现象都离不开它们。本文梳理了你需要掌握的练习 17A 的核心概念与技巧，提供了清晰的讲解以及与典型考试题目相匹配的范例。

1. Understanding Exponential Functions | 理解指数函数

An exponential function takes the form f(x) = a^x where the base a is a positive constant not equal to 1 (a > 0, a ≠ 1). The variable x sits in the exponent, which leads to rapid growth or decay.

指数函数的形式为 f(x) = a^x，其中底数 a 是一个大于 0 且不等于 1 的常数（a > 0, a ≠ 1）。变量 x 出现在指数位置，这使得函数表现出快速的增长或衰减。

The domain of any exponential function is all real numbers, while the range is (0, ∞). The graph always passes through (0, 1) because a⁰ = 1, and the x‑axis (y = 0) is a horizontal asymptote.

任何指数函数的定义域都是全体实数，而值域为 (0, ∞)。图像总是经过点 (0, 1)，因为 a⁰ = 1，并且 x 轴（y = 0）是一条水平渐近线。

When a > 1 the function is increasing (growth); when 0 < a < 1 it is decreasing (decay). Transformations such as vertical shifts y = a^x + c or horizontal shifts y = a^x-h are often tested.

当 a > 1 时函数递增（增长）；当 0 < a < 1 时函数递减（衰减）。像垂直平移 y = a^x + c 或水平平移 y = a^x-h 这样的变换经常出现在考题中。

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2. The Natural Exponential Function e^x | 自然指数函数 e^x

Among all exponential functions, the one with base e ≈ 2.71828 (Euler’s number) is the most important. It is written as e^x or exp(x). The function e^x is its own derivative, a property that makes it central to calculus and continuous growth models.

在所有指数函数中，以 e ≈ 2.71828（欧拉数）为底的函数最为重要。它记作 e^x 或 exp(x)。函数 e^x 的导数等于它本身，这一性质使其在微积分和连续增长模型中占据核心地位。

In IB Mathematics, you will meet e^x when studying exponential growth and decay, differential equations, and the natural logarithm. Its graph has the same shape as other exponential functions with base greater than 1, but the slope at x = 0 equals 1, making it unique.

在 IB 数学中，你会在学习指数增长和衰减、微分方程以及自然对数时遇到 e^x。它的图像形状与其他底数大于 1 的指数函数相同，但在 x = 0 处的斜率等于 1，这一点独一无二。

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3. Introduction to Logarithms | 对数入门

A logarithm answers the question: “To what power must the base be raised to obtain a given number?” If a^x = b, then x = log_ab, where a is the base, b is the argument (or antilogarithm), and x is the logarithm.

对数回答这样一个问题：“底数需要被提升到几次方才得到给定的数？”若 a^x = b，则 x = log_ab，其中 a 是底数，b 为真数，x 为对数。

Common logarithms use base 10, often written simply as log b. Natural logarithms use base e and are written as ln b. Remember: log_a1 = 0 and log_aa = 1 for any admissible base a.

常用对数以 10 为底，常简写为 log b。自然对数以 e 为底，记作 ln b。请记住：对任意合法的底数 a，总有 log_a1 = 0 以及 log_aa = 1。

The logarithmic function y = log_ax is the inverse of y = a^x. Its domain is (0, ∞), its range is ℝ, and its graph is the reflection of the exponential graph across the line y = x.

对数函数 y = log_ax 是 y = a^x 的反函数。它的定义域为 (0, ∞)，值域为 ℝ，其图像是指数函数图像关于直线 y = x 的反射。

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4. Laws of Logarithms | 对数运算法则

The three fundamental laws (valid for a > 0, a ≠ 1, M > 0, N > 0) allow you to simplify and manipulate logarithmic expressions:

三大基本法则（对 a > 0, a ≠ 1, M > 0, N > 0 成立）让你能够化简和处理对数表达式：

• Product law: log_a(MN) = log_aM + log_aN
• Quotient law: log_a(M/N) = log_aM − log_aN
• Power law: log_a(M^p) = p · log_aM

• 积的法则： log_a(MN) = log_aM + log_aN
• 商的法则： log_a(M/N) = log_aM − log_aN
• 幂的法则： log_a(M^p) = p · log_aM

It is essential to practise applying these laws in both directions – combining logs into a single logarithm or expanding a single log into sums and differences. Many IB problems test exactly this skill.

熟练地双向应用这些法则至关重要——涉及将对数合并为单个对数，或将一个对数展开为和与差。许多 IB 题目正是考察这一技能。

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5. Solving Exponential Equations | 解指数方程

There are two primary strategies for solving exponential equations: making the bases the same or taking logarithms of both sides. If you can express both sides as powers of the same base, you can equate the exponents directly.

解指数方程主要有两种策略：使底数相同，或者对两边取对数。如果能将两边表示为同一个底数的幂，就可以直接令指数相等。

For example, to solve 5^2x−1 = 125, recognise that 125 = 5³. Then 2x − 1 = 3, giving x = 2. Always check that your solution lies in the original domain.

例如，解 5^2x−1 = 125，注意到 125 = 5³，于是 2x − 1 = 3，解得 x = 2。务必检验解是否在原方程的定义域内。

When the bases cannot easily be made identical, take the natural logarithm (or log base 10) of both sides and use the power law to bring the exponent down. For instance, 3^x = 7 → x ln 3 = ln 7 → x = ln 7 / ln 3.

当底数不容易化为相同时，可两边取自然对数（或常用对数），并利用幂的法则将指数拉到前面。例如，3^x = 7 → x ln 3 = ln 7 → x = ln 7 / ln 3。

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6. Solving Logarithmic Equations | 解对数方程

Logarithmic equations often require using the laws of logs to condense expressions, followed by converting to exponential form. Always check that the arguments remain positive and that the bases are valid, as algebraic manipulations can introduce extraneous solutions.

对数方程通常需要利用对数法则将式子压缩，再转为指数形式求解。必须始终检查真数是否为正、底数是否有效，因为代数操作可能引入增根。

A typical approach is to combine log terms into a single logarithm, then write the equivalent exponential equation. For example, log₂(x) + log₂(x−2) = 3 becomes log₂(x(x−2)) = 3, so x(x−2) = 2³ = 8, leading to x = 4 or x = −2. The solution x = −2 must be rejected because log₂(−2) is undefined.

一个典型的解法是先把对数项合并成单个对数，再写出等价的指数方程。例如，log₂(x) + log₂(x−2) = 3 化为 log₂(x(x−2)) = 3，于是 x(x−2) = 2³ = 8，解得 x = 4 或 x = −2。解 x = −2 必须舍去，因为 log₂(−2) 无定义。

When solving equations like ln(5x+2) = 4, convert directly to exponential form: 5x+2 = e⁴, then solve for x. Remember that ln and e are inverse operations.

解如 ln(5x+2) = 4 的方程时，直接转为指数形式：5x+2 = e⁴，然后解出 x。请记住 ln 和 e 互为逆运算。

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7. Change of Base Formula | 换底公式

Calculators typically only provide keys for log₁₀ and ln. The change‑of‑base formula allows you to evaluate log_ab for any base a using either common or natural logs:

计算器通常只提供 log₁₀ 和 ln 键。换底公式允许你使用常用对数或自然对数计算任意底数的对数：

log_ab = log_cb / log_ca

In most cases, c = 10 or c = e. For example, to find log₅20, compute log 20 / log 5 ≈ 1.3010 / 0.6990 ≈ 1.861. This formula is also useful for solving exponential equations where the base is unfamiliar.

多数情况下，c = 10 或 c = e。例如，计算 log₅20，可用 log 20 / log 5 ≈ 1.3010 / 0.6990 ≈ 1.861。换底公式在底数不熟悉的指数方程中也很有用。

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8. Applications: Growth and Decay | 应用：增长与衰减

Real‑life situations such as population growth, radioactive decay, and compound interest are modelled by exponential functions of the form A = A₀ e^kt (continuous) or A = A₀ b^t. The sign of k determines growth (k > 0) or decay (k < 0).

现实生活中的情形，如人口增长、放射性衰变和复利，都可以用形如 A =

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